Detecting the most probable transition pathway based on optimal control theory

Many natural systems exhibit transitions when external environmental conditions (such as complex noise) spark a shift to a new and sometimes quite different state. Therefore, detecting the most probable transition pathway between metastable states of a stochastic dynamical system is a significant topic. The most probable transition pathway can be treated as the minimizer of the associated Onsager-Machlup action functional. We convert this variational problem for computing the most probable transition pathway into a deterministic optimal control problem. One traditional approach for an optimal control problem is via Pontryagin's Maximum Principle, but it is challenging in high dimensional systems. In this paper, we devise a method to detect the most probable transition pathway for stochastic dynamical systems, by combining Pontryagin's Maximum Principle with a successive approximation scheme and a nested neural network technique. We validate our method with three stochastic dynamical systems, including a double well system, a Maier-Stein system, and a Nutrient-Phytoplankton-Zooplankton system. Specifically, in order to illustrate its effectiveness, we test the error and convergence of our method with an upper bound of training loss. Our work contributes to a better understanding of transition phenomena in complex systems under random fluctuations.